Extensions of the representation modules of a prime order group

Hopf Galois structures on separable field extensions of odd prime power degree

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},μ)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper, we present several results on Hopf Galois structures on odd prime power degree separable field extensions. We prove that if a separable field extension of odd prime power degree has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. We determine the number of Hopf Galois structures of cyclic type on a separable field extension of degree $p^n$, $p$ an odd prime, such that the G…

素幂阶群扩张分类的轨道

Orbits classifying extensions of prime power order groups

本文研究有限阿贝尔p-群被p阶循环因子扩张的强同构类，通过第二上同调群上的轨道参数化，并计算初始p-群由至多3个元素生成时轨道的数目与大小。

On fibred biset functors with fibres of order prime and four

This note has two purposes: First, to present a counterexample to a conjecture parametrizing the simple modules over Green biset functors, appearing in an author's previous article. This parametrization fails for the monomial Burnside ring over a cyclic group of order four. Second, to classify the simple modules for the monomial Burnside ring over a group of prime order, for which the above-mentioned parametrization holds.

Hopf Galois structures on extensions of degree twice an odd prime square and their associated left braces

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

On the group of automorphisms of the Brandt $λ^0$-extension of a monoid with zero

The group of automorphisms of the Brandt $λ^0$-extension $B^0_λ(S)$ of an arbitrary monoid $S$ with zero is described. In particular we show that the group of automorphisms $\mathbf{Aut}(B_λ^0(S))$ of $B_λ^0(S)$ is isomorphic to a homomorphic image of the group defines on the Cartesian product $\mathscr{S}_λ\times \mathbf{Aut}(S)\times H_1^λ$ with the following binary operation: \begin{equation*} [φ,h,u]\cdot[φ^{\prime},h^{\prime},u^{\prime}]= [φφ^{\prime},hh^{\prime},φu^{\prime}\cdot uh^{\prime}], \end{equation*} where $\mathscr{S}_λ$ is the group of all bijections of the cardinal $λ$, $\math…

P. Hall 关于极大子群指数定理的推广

Extensions of a theorem of P. Hall on indexes of maximal subgroups

本文推广 P. Hall 关于极大子群指数与有限群可解性的经典定理，证明在极大子群幂零或真非极大子群含于素数（平方）指数子群时，$G$ 仍可解。